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2 Moblty of Spatal Paallel Manpulatos Jng-Shan Zhao ule hu and Zh-Jng eng Depatment of Pecson Instuments and Mechanology, snghua Unvesty, Bejng 84, P. R. hna Open Access Database Intoducton hs chapte focuses on the moblty analyss of spatal paallel manpulatos. It fst develops an analytcal methodology to nvestgate the nstantaneous degee of feedom (DO) of the end-effecto of a paallel manpulato. And then, the nstantaneous contollablty of the end-effecto s dscussed fom the vewpont of the possble actuaton schemes whch wll be especally useful fo the desgnes of the paallel manpulatos. Va compang the dffeences and essental moblty of a set of undeactuated, ove actuated and equally actuated manpulatos, ths chapte demonstates that the undeactuated, ove actuated and equally actuated manpulatos ae all substantally fully actuated mechansms. hs wok s sgnfcantly mpotant fo a desgne to contve hs o he manpulatos wth undeactuated o ove actuated stuctues. Based on the analytcal model of the DO of a spatal paallel manpulato, ths chapte develops a geneal pocess to synthesze the manpulatos wth the specfed moblty. he outstandng chaactestcs of the synthess method ae that the whole pocess s also analytcal and each step can be pogammed at a compute. Because of the estctons of the tadtonal geneal moblty fomulas fo spatal mechansms, a lot of mechansms havng specal manoeuvablty mght not be syntheszed. Howeve, any mechansm can be syntheszed wth ths analytcal theoy of degees of feedom fo spatal mechansms.. he vald means to nvestgate the moblty of a mechansm he quck calculaton appoaches based on the algeba summatons of the numbe of the lnks, jonts and the constants nduced by the jonts can not be completely pefected by tself. hs s tue even the analytcal methods ae appled n seekng the common constants (Hunt, 978)(Waldon, 966)(Huang, 6). hese poblems ae becomng moe and moe obvous wth the advent of spatal paallel manpulatos. he pmay consdeatons of the desgnes fo the paallel manpulatos have been focused on nothng but the moblty of the end-effecto and ts contollablty. heefoe, the concept of geneal moblty of a mechansm should be dvded nto two basc concepts the degee of feedom of the end-effecto and the numbe of actuatons needed to contol the end-effecto. Wth ths egad, ths chapte fst ntoduces two pmay defntons: Souce: Paallel Manpulatos, owads New Applcatons, Book edted by: Huapeng Wu, ISBN , pp. 56, Apl 8, I-ech Educaton and Publshng, Venna, Austa

3 468 Paallel Manpulatos, owads New Applcatons Defnton : he DO of an end-effecto totally chaactezes the motons of the end-effecto ncludng the numbe, type and decton of the ndependent motons (Zhao et al, 4a)(Zhao et al, 6a). Defnton : he confguaton degee of feedom (DO) of a mechansm wth an end-effecto ndcates the ndependent numbe of actuatons equed to unquely contol the end-effecto unde a confguaton (Zhao et al, 4b)(Zhao et al, 6c). Obvously, the DO of an end-effecto n numbe s not lage than 6 but the ndependent numbe of actuatons equed to unquely contol the end-effecto mght be any nonnegatve ntege. Beang the above two defntons n mnd, one can fall nto two steps to nvestgate the moblty of a mechansm the DO of the end-effecto and the DO of the mechansm wth the pescbed end-effecto. he fome defnton ndcates the full nstantaneous moblty popetes of the end-effecto though a mathematcs concept of fee moblty space whle the late one pesents the nstantaneous contollablty of the mechansm system. By defnton, one can fnd that the DO of an end-effecto s only subjected to the constant(s) exeted by the knematc chan(s) connectng the end-effecto wth the fxed base o gound. Besdes, the degee of feedom of the end-effecto, nstantaneously assocated wth the spatal confguatons of the knematc chan(s), should clealy depct the numbe, the decton and the type of the fee moton of the end-effecto nstantaneously. heefoe, only analytcal methods can fulfl such a task. Afte obtanng the fee motons of the end-effecto, an engneeng queston wll natually ase how many actuatons ae needed to contol the end-effecto? By defnton, one can fnd that a checkng pocess s gven fo vefyng the contollablty of the mechansm wth the specfed end-effecto. Besdes, ths pocess can also allow the dffeent selectons of the actuaton schemes, whch s most adapted to the concept desgn of a manpulato. onsequently, the vald means to nvestgate the moblty of mechansms can be addessed as: () nvestgate the nstantaneous DO of the pescbed end-effecto; and () nvestgate the numbe of actuatons equed to unquely contol the end-effecto of the mechansm. o the nstantaneous chaactestcs of the moblty of a mechansm, only analytcal means s acceptable fo such a task. Because of the elegance n depctng the elatonshp between the motons and the constants, ecpocal scew theoy does be a well selecton to accomplsh the task. heefoe, the followng analytcal model fo the moblty of a paallel manpulato wll be bult up by applyng the ecpocal scew theoy. Accodng to ecpocal scew theoy (Hunt, 978)(Phllps, 984)(Phllps, 99)(Phllps et al, 964)(Waldon, 966)(Ball, 9), a scew $ s defned by a staght lne wth an assocated ptch h and s convenently denoted by sx Plücke homogeneous coodnates: s $ = s hs + whee s denotes decton atos pontng along the scew axs, s = s defnes the moment of the scew axs about the ogn of the coodnate system, s the poston vecto of any pont on the scew axs wth espect to the coodnate system. onsequently, the s scew axs can be denoted by the Plücke homogeneous coodnates $ = axs s. ()

4 Moblty of Spatal Paallel Manpulatos 469 Assume s = s+ ( L M N ) h s = ( P Q R) () s and pesumng s, one obtans the nstant onsdeng s ( s + h s) = s s + s h = h ptch of a scew: s h = ( s + hs) s LP + MQ + NR = L + M + N heefoe, the axs of the scew can also be denoted as: axs ( L M N P Lh Q Mh R Nh) $ = (4) Assume that the vecto of the pojectve pont of the ogn on the scew axs s epesented by, thee wll be s OP and: O P Accodng to equatons () and (), thee ae: ( O s) = ( s s) ( ) P O s P O s = s P OP s (5) (3) s = OP ( L M N ) s = ( P Lh Q Mh R Nh) (6) whch yelds: M ( ) ( R Nh) N( Q Mh) s s OP ( ) ( ) ( ) ( ) O = = N P Lh LR Nh P L + M + N (7) s LQ Mh M P Lh onsequently, f the Plücke coodnates of a scew ae gven, one can easly obtan the unt decton vecto, s, the ptch, h, the scew axs and the vecto of the pojectve pont of the ogn on the axs, O P, wth equatons () though (7). If the ptch of a scew equals zeo, the scew coodnates educe to be: s $ = (8) s whch s just the Plücke homogeneous coodnates of the scew axs. In fact, fomula (8) unquely defnes a lne n a thee-dmensonal space. Assume that pont O P s the pojectve pont of the ogn on a lne l and pont A s any othe pont on the lne. hen,

5 47 Paallel Manpulatos, owads New Applcatons a s = O + P OPA = O (9) P s A + whee s s a decton vecto of lne l, a s the length of lne segment he moment of lne l about the ogn at pont A wll be: O P A. s s s a = A = + s = s O P s O P () om equatons (9) and (), one obtans that the moment of a lne about the ogn s elevant to the pont s selecton on the lne. If a scew passes though the ogn of the coodnate system, the scew coodnates can be denoted as: s $ = h s On the othe hand, f the ptch of a scew s nfnte, the scew s defned as: () whee ( ) $ = s = s a thee dmensonal vecto. Accodng to the above defntons, a scew assocated wth a evolute pa s a twst of zeo ptch pontng along the pa axs whle a scew assocated wth a psmatc pa s a twst of nfnte ptch pontng n the decton of the tanslatonal gude lne of the pa. om equaton (), one has known that the knematc scew s often denoted n the fom of Plücke homogeneous coodnates: ( L M N P Q R) () $ = (3) whee the fst thee components denote the angula velocty, the last thee components denote the lnea velocty of a pont n the gd body that s nstantaneously concdent wth the ogn of the coodnate system. Smlaly, $ s defned as: L M N P Q R $ = (4) whee the fst thee components denote the esultant foce and the last thee components denote the esultant moment about the ogn of the coodnate system. wo scews, $ and $, ae called to be ecpocal f they satsfy the equaton: LP + MQ + NR + PL + QM + RN = (5)

6 Moblty of Spatal Paallel Manpulatos 47 Obvously, the fee motons (geneal twsts) $ and the pescbed constants (geneal wenches) $ of an equlbum gd body should satsfy equaton (5). Equaton (5) s often wtten fo shot (Kuma, 99): ( ) E$ = $ (6) 3 I3 whee $ and $ ae column vectos, E =, and I 3 and 3 ae 3 3 dentty and I3 3 zeo matces, espectvely. Smlaly, f one gets a set of temnal constants exeted to a gd body, ts fee moton(s) can also be solved though equaton (6). Next, one can nvestgate the nstantaneous moblty of the end-effecto of a paallel manpulato wth equaton (6).. he degee of feedom of the end-effecto of a paallel manpulato he fee motons of the end-effecto can be nstantaneously expessed n a set of Plücke homogeneous coodnates n one atesan coodnate system. he man steps ae:. Investgate the emnal onstants of the Knematc hans nn knematc chans In geneal, any paallel manpulato can be decomposed nto ( ) connectng the end effecto wth the base. In ode to nstantaneously analyze the moblty popetes of the end-effecto, ths secton only establshes one absolute coodnate system. Afte establshng the coodnate system, the Plücke homogeneous coodnates of all knematc pas n a chan can be obtaned. Goup all of the knematc scews of the same chan to be $ (,, n) = and solve the temnal constant(s), $ wth equaton (6). In fact, f all of the temnal constants of the knematc chans ae ganed, the constants exeted to the end-effecto, denoted by $ E, should also be obtaned. he dmenson of constant spaces spanned by the temnal constants of knematc chans can be smplfed as d = Rank $ E.. Solve the ee Moton(s), $ E, of the End-Effecto wth Equaton (6) Natually, the moblty popetes of the end-effecto s fully expessed by DO can be expessed as: $ E. Its numbe of M Rank = $ E = 6 d (7) Now, the DO of the end-effecto of the paallel manpulato shown n g. can be nstantaneously nvestgated wth the above two steps. In ths manpulato, the end-effecto has thee dentcal PPRR knematc chans connected wth the fxed base. o the 3 sake of modellng, one can establsh any atesan coodnate system fo the manpulato. A =,, 3 s denoted by Assume that the decton vecto of the psmatc jont ( ) = ( a b c ), the decton vecto of the psmatc jont (,, 3) ea B = s denoted by

7 47 Paallel Manpulatos, owads New Applcatons ( a b c ) =, the otatonal vecto of the evolute jont B s denoted by eb e B A B ( b c b c a c a c a b a b ) = e e =, the otatonal vecto of the evolute jont (,, 3) = s denoted by e = e. Also suppose that e B A e e and ee =. A A3 B A g. a 3-PPRR Spatal Mechansm So, the knematc scews fo each knematc chan can be expessed as: $ A = B $ A $ B $ $ (8) B whee $ = ( a b c ), = ( a b c ) A $, B $ B = bc bc ac ac ab ab yabab ( ) zbcbc ( ) xacac ( ) B B B, zacac ( ) xabab B ( ) ybcbc ( ) B B $ = bc bc ac ac ab ab yabab ( ) zbcbc ( ) xacac ( ). zacac ( ) xabab ( ) ybcbc ( ) he temnal constants of the knematc chan can be solved wth (6): 3 $ = AB $ $ $ (9) whee $ = ( b c ), = ( b c ) a $, a $ 3 = b c b c a c a c a b a b y z ( ab ab ) ( a c a c ) z x ( bc b c) ( a b a b ) x y ( ac ac ) ( b c b c ).

8 Moblty of Spatal Paallel Manpulatos 473 Accodng to the mechansm shown n g., e = e = e. heefoe, the temnal B B B3 constants exeted to the end-effecto by these thee knematc chans ae: It s not dffcult to fnd that the ank of $ = $ 3 $ $ 3 $ $ $ $ 3 () $ expessed by equaton () s 5, and the fee 3 motons of the end-effecto 3 can be agan solved wth equaton (6): ( abc ) $ = () 3 3 Equaton () ndcates that the end-effecto has one tanslatonal DO along the decton vecto eabc ( ) 3 =. Of couse, the numbe of the DO of the end-effecto s = eabc = and the type s tanslaton, whch s 3 3 fully epesented by the scew expesson (). Rank $, the decton s ( ). he numbe of actuatons equed to contol the end-effecto of a spatal paallel manpulato Afte obtanng the nstantaneous moblty of the end-effecto, one can dectly exet M actuatons to the manpulato, and then nvestgate the DO of the end-effecto by solvng the fee moton(s) of the end-effecto wthn ts wokspace. If the newly solved moton(s), n denoted by $, =,, E n E, satsfy that ( ) $, then, addtonal actuatons ae needed unde ths confguaton and the actuaton scheme. Of couse, we can ethe eselect the actuaton scheme o add E = n untl ( ) n Rank $ E moe actuaton(s) unde ths confguaton $. he total numbe of actuatons unde the confguaton wth ths actuaton scheme s the DO. Howeve, what must be ponted out s that the actuaton(s) should not be exeted to the jont when the newly nceased temnal constant can be tansfomed by the othe actuaton(s). Othewse, the ove constant case wll occu. When thee ae a lot of possble actuaton schemes any one of whch can be selected to set the actuatos, the contollablty of the manpulato s also affected by the actuaton scheme s selecton. o nstance, one can analyze the numbe of actuaton(s) equed to contol the end-effecto of the paallel manpulato shown n g.. Because the numbe of DO of the end-effecto s, t s easonable fo us to expect that the end-effecto can be fully contolled only wth one actuaton. If one actuaton s exeted to any jont of the mechansm, A fo an example, t s not dffcult to fnd that the end-effecto stll emans one tanslatonal DO n the decton ( a b c ) e = when one epeats the above two steps n secton.. heefoe, one has to add anothe actuaton to the mechansm. Of couse, he can add the second actuaton to any one of the est jonts. Howeve, t s not dffcult to pove that the

9 474 Paallel Manpulatos, owads New Applcatons end-effecto wll not be contolled unless the second actuaton s exeted to the psmatc jont B ( =,, 3) unde the condton that the fst actuaton s exeted to ( =,, 3) A. Howeve, just as mentoned above, the new-added actuaton should not be accepted f the newly-nceased temnal constant can be obtaned by tanslatng the fome actuaton(s). o an example, f the second actuaton s assgned to the evolute jont B, the newlynceased temnal constants of the knematc chan A B wll be: ( a b c y c z b z a x c x b y a ) $ = () n Equaton () s the tansfomaton of the actuaton exeted to the psmatc jont A. So, the newly-added actuaton s an ove actuaton fo the actuaton scheme whose fst actuaton s assgned to A. Of couse, one can also exet the second actuaton to the psmatc jont A afte assgnng the fst actuaton to the psmatc jont A. Agan, one can fnd that the end-effecto stll has the fee tanslaton n the decton eabc ( ) 3 = when one epeats the above two steps n secton.. So, one can contnue to add the thd actuaton to the psmatc jont A 3. Howeve, the end-effecto wll not be contolled untl a fouth actuaton s appled to one of the psmatc jonts, B, B and B 3. hs foms a second actuaton scheme. So, unde ths actuaton scheme, the numbe of actuatons needed to contol the end-effecto shown n g. s 4. he dffeences between the second actuaton scheme and the fst one ae that the second one not only completely contol the end-effecto but also completely contol evey lnk n the manpulato. he selectons of dffeent actuaton schemes can be well accomplshed by a compute especally when the possble selectons ae numeous such as the one shown n g.. Unfotunately, ths popetes of a mechansm s gnoed by the geneal moblty fomulas. 3. he substantal moblty of undeactuated, ove actuated and equally actuated manpulatos A manpulato s sad to be undeactuated when the numbe of actuatos n the manpulato s smalle than the numbe of degees of feedom of the mechansm (Lalbeté & Gosseln, 998). When appled to mechancal fnges, the concept of undeactuaton leads to shape adaptaton,.e. undeactuated fnges wll envelope the objects to be gasped and adapt to the shape although each of the fnges s contolled by a educed numbe of actuatos (Lalbeté & Gosseln, 998). he concept of undeactuaton n obotc fnges wth fewe actuatos than the degees of feedom allows the hand to adjust tself to an egulaly shaped object wthout complex contol stategy and sensos (Bglen & Gosseln, 6a). hese undeactuated manpulatos ase n a numbe of mpotant applcatons such as space obots, hype edundant manpulatos, manpulatos wth stuctual flexblty, etc (Jan & Rodguez, 993). he fact that the undeactuated obotc fnges allow the hand to adjust tself to an egulaly shaped object makes t possble that no complex contol stategy o numeous sensos ae necessay n these manpulatos (Bglen & Gosseln,

10 Moblty of Spatal Paallel Manpulatos 475 6b). Howeve, the ove actuated mechancal systems often occu n bomechancal systems dung the contact wth gound and s ecently ntoduced n edundantly actuated paallel obots. Y and Km (Y & Km, ) desgned a sngulaty fee load-dstbuton scheme fo a edundantly actuated thee-wheeled Omndectonal moble obot. he most outstandng advantage of the edundantly actuated moble obot s that the sngulates of the mechansm can be well avoded. Yu and L (Yu & L, 3) nvestgated the tajectoy geneaton fo an ove actuated paallel manpulato, n whch thee s one edundant actuato. Of couse, the edundant actuato(s) and the equed actuato(s) must obey a cetan elatonshp detemned by the mechansm, whch wll be dscussed n secton 3.. hs secton ams at clafyng the substantal elatonshps between the undeactuated, ove actuated and the equally actuated manpulatos. he undeactuated manpulato, whch s also called unde-detemnate nput system, means that the numbe of actuatons povded s less than that s necessay; whle the ove actuated manpulato, whch s also called edundant actuaton o edundant nput system, means that the numbe of actuatons povded s lage than that s necessay. Equally actuated manpulato, whch s also called fully actuated o detemnate system, means that the actuatons povded s equal to that s needed. om the vewpont of mechansms, ths classfcaton of manpulatos seems to be easonable and has been wdely used n engneeng. Howeve, t s not a popely scentfc categozaton fo mechansms. heefoe, ths secton wll befly study the substantal elatonshps between the undeactuated, ove actuated and equally actuated manpulatos that ae easly msundestood n engneeng applcatons. 3. he essence of the undeactuated manpulato o begn wth ths secton, one mght fst nvestgate a famous nveted pendulum system shown n g., whch s also a epesentatve, undeactuated mechancal system. hs nveted pendulum system s a plana two degees of feedom catenaton mechancal system. he vehcle can only make ecpocal tanslaton along the x -axs and the pendulum can only otate about the pvot attached to the movng vehcle. In applcatons, only one actuaton s povded to contol the system, whch seems to conflct wth the defnton of a fully actuated mechansm. In ode to eveal the essence of ths puzzlng phenomenon, one mght fst tun to analyze the dynamcs of ths two-degee-offeedom system. Suppose the mass of the vehcle s denoted by M, the mass of the pendulum s m and the dstance fom the pvot attached to the vehcle to the mass cente of the pendulum s l and the moment of neta of the pendulum s denoted by J. he dynamcs of the system can be mmedately establshed va Lagange method. he knetc enegy of the vehcle s: v = Mx whee v epesents the knetc enegy of the vehcle. he knetc enegy of the pendulum s: p d d = m ( sn ) + ( cos ) + x + l θ m l θ J θ dt dt whee p epesents the knetc enegy of the pendulum.

11 Paallel Manpulatos, owads New Applcatons 476 g. a sngle nveted pendulum system he total knetc enegy of the system s: ( ) cos = + = θ θ θ ml J x ml x m M p v he potental enegy of the system s: mglcosθ V = heefoe, the Lagange functon of the system s: ( ) θ θ θ θ cos cos mgl ml J x ml x m M L = (3) whee L ndcates the Lagange functon. he dynamcs equatons fo the two-degee-of-feedom system shown n g. can be expessed as: = = τ θ θ L L dt d x L x L dt d (4) whee τ epesents the toque exeted to the evolute jont that connect the nvese pendulum and the vehcle. x θ l M m J

12 Moblty of Spatal Paallel Manpulatos 477 om the Lagange functon, one mmedately obtans: = ( M + m) x+ mlθ cosθ L x = mlx cosθ + J + ml θ L θ L, = x L, = ml θ snθ θ g x Substtutng the above equatons nto equaton (4), one has: ( M + m) x mlθ snθ + ml θ cosθ = θ + mlx cos = J + ml θ mglsnθ τ Of couse, n the undeactuated condton, thee s τ =. he fst fomula n equaton set (5) s the appaent actuaton fomula whle the second one n equaton set (5) s a hdden elatonshp of the mechancal system, n whch the gavty, the neta foce and moment of the pendulum ae assocated pecsely. As a matte of fact, theefoe, ths elatonshp depcted by the second fomula n equaton set (5) povded anothe actuaton constant fo the two-degees-of-feedom mechancal system n dynamcs but not n statcs. heefoe, the mechancal system shown n g. s fully actuated n dynamcs but not n statcs. When x = and τ =, equaton set (5) can be smplfed as: (5) θ = ± mgl cosθ J + ml mlsnθ (6) whee ± s detemned by the ntal condton of the system and the sgn should be + n the case shown n g.. om equatons (5) and (6), t s not dffculty to fnd that the nveted pendulum system shown n g. can only keep a dynamc equlbum but not a statc equlbum whch s a pmay equement fo a mechansm. A much moe famla example s the dffeental gea tan mechansm used n the dvng axle of all knds of automobles. he basc mechansm stuctue s shown n g. 3. he pnon gea tansfoms the toque fom the engne to the dvng axle shafts by a dffeental gea tan mechansm, n whch the ng gea shown n g. 3 acts as an actuato and the ght and the left shafts act as executos. Obvously, ths mechansm also has two degees of feedom. Howeve, the actuaton s just one otatonal nput fom the pnon gea. One mght daw a concluson n haste that ths mechansm should be an outstandng epesentatve example fo the applcatons of undeactuated mechancal systems because t s so wdely used n the moden vehcles. hs mechancal system s eally qute dffeent fom the nveted pendulum system shown n g. n that the hdden mechancal constant o actuaton s moe easly gnoed. he eacton dffeence between the ght and left wheels fom the oad suface povdes such an

13 478 Paallel Manpulatos, owads New Applcatons actuaton, whch s appaent when the eactons to the ght and left wheels fom the oad suface ae dffeent, and whch often occus when the vehcle makes a ght o left tun. g. 3 the dffeental gea tan mechansm Anothe faclty usually used n cvl engneeng s the netal amme shown n g. 4. hs can also be modelled wth a plana mechansm shown n g. 5. he appaent actuaton s povded by the eccentc foce of the eccentc oto unde the actuaton of the electc moto. Howeve, the moton of the amme s body s ndetemnate f the contol of h s not exeted to the handle. heefoe, the neta amme s not an undeactuated mechancal system but a fully actuated system although the appaent actuaton seems to be estcted to the eccentc foce esultng fom the eccentc otatng oto. g. 4 the neta amme g. 5 the mechansm of the neta amme om the above analyss, t s not dffcult to fnd that all the undeactuated mechancal systems ae substantally actuaton detemnate fom the vewpont of mechansms.

14 Moblty of Spatal Paallel Manpulatos he essence of the ove actuated manpulato Ove-detemnate actuaton manpulatos also wtnesses wde applcatons n mechancal engneeng, especally n bomechancal engneeng. In ode to nvestgate the essence of these manpulatos, ths secton addesses ths poblem va some mechansm examples. As a smple example, one mght fst nvestgate the moton of a vehcle wth one degee of feedom unde the actons of two pesons shown n g. 6. he vehcle can only tanslate fowad and backwad along the oad decton. Howeve, two dffeent actons ae exeted to both sdes of the vehcle. So, t s an ove actuated mechancal system. g. 6 an ove actuated mechancal system x x x Out of queston, the vehcle shown n g. 6 wll move along the decton of the esultant foce of the two pesons, n spte of whch the two actuatons ae not ndependent because these two actuatons should satsfy that x = x = x. Othewse, the two actuatons mght not do contnuous wok to the vehcle. hese addtonal constants ae also called complant equatons. y l 4 B l 3 A(o) l α l β x g. 7 a plana fou-ba mechansm wth two actuatons Next, one can consde a plana fou-ba mechansm unde two actuatons shown n fgue 7. Obvously, only one actuaton s needed to contol the mechansm. In engneeng applcatons, howeve, t s also avalable to exet two actuatons to ncease the nput toque o foce to dve the mechansm to output a lage powe. heefoe, the mechansm n such a case s a epesentatve of the ove actuated mechancal systems. D

15 48 Paallel Manpulatos, owads New Applcatons ollowng, one can nvestgate the poblems that mght be gnoed o msundestood. o the sake of convenences, a coodnate system s establshed by settng the ogn to supempose wth evolute jont A and x -axs along the lnk AD and y -axs pependcula upwad to lnk AD. If the plana fou-ba mechansm has a detemnate moton, the equaton below should hold: ( cos + l l cosα ) + ( l snβ l α ) l β = l (7) 3 3 sn heefoe, dffeentatng equaton (7) wth espect to tme and eaangng yelds: 4 β l = l3 α [ lsn α + l3sn( α β )] [ l snβ + l α β )] (8) whee α and β epesent the angula veloctes of the cank AB and the ocke D shown n g. 7, ndvdually. heefoe, the actuatons exeted to the cank AB and the ocke D should keep n a pecse elatonshp specfed by equaton (7). Othewse, the lnk B mght be cacked due to the nceasng ntenal foces. Equaton (7) o (8) s the complant equaton fo the ove actuated manpulato shown n g. 7. onsequently, t s not dffcult to fnd that thee always ae complant constant equatons fo the ove actuated mechancal systems. And theefoe, these mechancal systems ae also substantally equally actuated. 3.3 he poblems to be noted n engneeng applcatons he dextety of an undeactuated manpulato dffes fom the dextety of a fully actuated one, even f the mechancal stuctues ae dentcal. heefoe, the undeactuated mechancal systems ae wdely used n the cases fo fault toleance and enegy savng puposes. om the above analyss, one knows that any mechancal system that has a detemnate moton should be an equally actuated system n essence. Next, one nvestgates an undeactuated mechancal fnge wth etun actuaton shown n g. 8. hs mechansm s used n the fnge of the Unted States patent ntally appled by Gosseln et al (Gosseln & Lalbeté, 998) fo dextety hand n 998. he pmay stuctue of the mechansm shown n g. 8 (Bglen & Gosseln, 6a) s a plana fou-ba mechansm. Lnks AB and AD ae smultaneously pvoted wth the fxed wst. Lnks AB and B ae connected by a passve spng. Next, the moblty of the mechansm wll be nvestgated n seveal cases. stly, when the fnge does not contact any object, the lnks AB and B connected by a passve spng mght be dsposed as one lnk, and theefoe, ABDE foms one lnk and otates about the fxed pvot, A, unde the actuaton of the foce. When AB contacts a taget object, the lnk AB wll degeneate to an unmovable base attached to the wst, and theefoe, the spng wll be defom unde the acton of the foce and fnge ABDE foms a eal fou-ba mechansm. hs wll be holdng untl the sde BE also touches the bounday of the taget object, afte whch the contnuous nceasng of the foce wll only esults the defomng of the taget object.

16 Moblty of Spatal Paallel Manpulatos 48 E B E B A D D A g. 8 undeactuated mechancal fnge wth etun actuaton he above analyss ndcates that the so called undeactuated mechancal fnge s equally o fully actuated at any nstant fom the vewpont of mechansms. onsequently, no matte what knd does a mechansm belong to, t should have a detemnate moton and equal actuaton(s) at any nstant, whch should be patculaly notced n the concept desgn of undeactuated mechancal systems. heoetcal and example analyss ndcate that the undeactuated, ove actuated and fully actuated mechancal systems ae all substantally equally actuated mechansms. 4. Synthess of a spatal paallel manpulato wth a specfed moblty Usually, suspenson s a geneal tem of the equpments tansfomng foces and moments fom the wheel to the vehcle body. Its pmay functon s to detemne the geomety of the wheel moton dung jounce and ebound, and to wthstand foces and moments on the suspenson n acceleatng moton (Raghavan, 996). he de and handlng chaactestcs of a vehcle ae heavly dependent on the knematc and complance popetes of the suspenson mechansm (Raghavan, 5). ompaed wth dependent suspensons, ndependent suspensons can elmnate undesable dynamc phenomena such as shmmy and caste wobble esultng fom wheel couplng n sold-axle suspensons (Raghavan, 996). he most common ndependent suspenson mechansms utlzed n automobles ae shot-long am suspenson (Suh, 998), the MacPheson stut (Raghavan, 5), the multlnk suspensons (Smonescu, ), and the shot-long am font suspenson wth a tue kngpn (Muakam, 989), etc. Most automotve ndependent suspenson mechansms ae sngle degee-of-feedom mechansms wth the pedomnant moton beng wheel jounce and ebound. In ode to allow the wheel to pass the uneven tean wthout slppng, hakaboty and Ghosal (hakaboty & Ghosal, 4) nvestgated the knematcs of a wheeled moble obot movng on uneven tean by modelng the wheels as a tous and poposng a lateal passve jont. Applcatons ndcate that the wheel oentaton and

17 48 Paallel Manpulatos, owads New Applcatons poston paametes such as kngpn, caste, cambe, toe change, axes dstance, and the wheel tack ae pmay consdeaton n the desgn of suspenson mechansm. hese paametes, as a matte of fact, ae dependent on the wheel jounce and ebound, an ndependent paamete (Raghavan, 5). heefoe, a patcula gd gudance mechansm whose end-effecto only has one staght lne tanslaton should mantan the oentaton and poston paametes nvaable. Yan and Kuo (Yan & Kuo, 6) addessed the topologcal epesentatons and chaactestcs of vaable knematc jonts, whch mght be utlzed n spatal mechansm synthess. By consdeng wokspace, dextety, stffness and sngulaty avodance, Asenault and Boudeau (Asenault & Boudeau, 6) dscussed the synthess poblems of plana paallel mechansms. In the hstoy of mechansm synthess, a sgnfcant example s that the ceaton of lnkages to poduce exact staght lne moton was an mpotant engneeng as well as a mathematcal poblem of the 9th centuy (Kempe, 877). Whle many engnees and mathematcans wee seachng fo a 4-5- o 6-ba staght lne lnkage all suffeed fom the fact that they could not attan such a moton n the mddle of 9th centuy, Peaucelle nvestgated an eght ba lnkage shown n g. 9 and dscoveed he could geneate an exact staght lne moton fom a otay nput. g. 9 Stuctue of Peaucelle-Lpkn Eght-Ba Lnkage hs nventon was ecognzed by seveal mathematcans as beng vey mpotant to the desgn of geneal mathematcal calculatos (Kempe, 877). hs eght-lnk lnkage was the one of the fst to poduce exact staght lne moton and was ndependently nvented by a ench engnee named Peaucelle and by a Russan mathematcan named Lpkn (Kempe, 877), whch s theefoe often called Peaucelle-Lpkn eght-lnk lnkage. Howeve, Peaucelle-Lpkn lnkage s mostly utlzed as a moton geneato but not a gd gudance mechansm. Obvously, because of ts complexty, such a mechansm can not be used as a suspenson n spte of the fact that t can eally make the wheel move n a staght lne dung jounce and ebound. heefoe, ths secton fst dscusses the synthess pocesses wth the analytcal model of the nstantaneous moblty of a manpulato fo the gd gudance mechansm wth the specfed moblty; and then pesents a ectlnea moton geneatng manpulato that can be utlzed as a suspenson mechansm. he geneal synthess pocess mght be: Step : Expess the fee motons equed fo the pescbed end-effecto n Plücke coodnates at a atesan coodnate system.

18 Moblty of Spatal Paallel Manpulatos 483 he Plücke coodnates of the specfed motons should be fstly expessed n a atesan coodnate system. hs chapte supposes that the twsts of the fee moton(s) of the end- effecto ae denoted by $ End. Step : Solve the constant(s) exeted to the end-effecto by ts knematc chan(s). Accodng to ecpoctes between fee moton(s) and constant(s) of an end-effecto, the constant(s) appled to the end-effecto can be solved wth the equaton (6): $ End E$ End = (9) whee $ End ndcates the specfed fee moton(s) of the end-effecto and $ End denotes any constant appled to the end-effecto. Step 3: Decde the numbe of knematc chans, m ( m ), that wll be used to connect the end-effecto wth the fxed base. If evey lnk n the chan s connected to at least two othe lnks, the chan foms one o moe closed loops and s called a closed knematc chan; f not, the chan s efeed to as open (Shgley & Ucke, 98). o the late open chan case, the synthess s smply stated as: any knematc chan s feasble f the twst bass, $ B, of the chan contans followng steps should be futhe dscussed f the mechansm s a closed one. Step 4: Synthesze the temnal constant(s) of each knematc chan. Suppose $ End. Howeve, the = $ $ End $ End n End $ End (3) whee n ndcates the dmenson of the constant bass of the end-effecto. Suppose that the temnal constant(s) of the th ( =,,, m ) knematc chan s denoted by $, the temnal constant(s) of the chan mght be syntheszed wth: whee K [ K K K ] n End =, and ( ) K $ = $ K (3) j = k k k and j =,,,n. o a feasble mechansm that makes the end-effecto only have the pescbed fee moton(s), the necessay and suffcent cteon s that the esultant temnal constant(s) of all these m knematc chan(s), m U $ = n, should be equvalent to $ End. hs s called the constucton cteon of the feasble knematc chans. he necessty and suffcency of ths cteon can be mmedately deduced fom equaton (9) wth lnea algeba theoy. Step 5: Solve the twst bass of the th knematc chan wth the temnal constants, $, syntheszed n step 4.

19 484 Paallel Manpulatos, owads New Applcatons Wth ecpocal scew theoy, a bass of the twst(s) of the th knematc chan, denoted by $ B, can be obtaned by solvng the followng equaton: $ E$ B = (3) whee $ epesents the temnal constant(s) of the th knematc chan syntheszed n step 4. Step 6: Synthesze the twst(s) of the th knematc chan wth the twst bass of the th chan, $ B, obtaned n step 5. Suppose $ = $ B $ B B $ B n (33) whee n ndcates the dmenson of the twst bass of the th ( =,,, m ) knematc chan. Accodng to lnea algeba, any twst of the th knematc chan can be expessed as the lnea combnatons of the twst bass of the chan: = c c. c n whee ( ) B a $ = $ (34) onsequently, the twsts of each knematc chan can be syntheszed though equaton (34). Howeve, n ode to keep the twsts of the th chan to be equvalent to the twst bass of the chan, the ank of the total twsts syntheszed though equaton (34) should equal the dmenson of the twst bass of the chan. hs s called the constucton cteon of the feasble knematc chans. he necessay and suffcent of ths cteon can be mmedately obtaned fom equaton (3). Accodng to the constucton ctea and, the equed synthess taget of a mechansm can be gadually accomplshed wth the above sx steps. Obvously, wth these sx steps, dffeent peson mght synthesze dffeent knematc chans and dffeent mechansms. Howeve, all the end-effectos of the mechansms syntheszed wth the same ctea wll suely have the dentcal specfed fee moton(s). he next secton wll apply these steps to synthesze a gd gudance mechansm that can be utlzed as a suspenson of an automoble. he synthess taget now s to use the least numbe of lnks and pue evolute jonts to desgn a mechansm whose end-effecto has one pue tanslaton along an exact staght lne; theefoe, the mechansm must be a closed one. he eason s that t wll need at least two actuatons to geneate a pue staght lne tanslaton wth an open chan mechansm. And theefoe, fo the pupose of the suspenson equed, one at least needs two knematc chans to geneate a pue staght lne tanslaton wth one actuaton nput. Accodng to step, the specfed fee moton of the end-effecto should be expessed n a atesan

20 Moblty of Spatal Paallel Manpulatos 485 coodnate system. Wthout loss of genealty, the pecse staght lne tanslaton of the endeffecto can be assumed to paallel z -axs. heefoe, the fee moton can be descbed n Plücke coodnates as: ( ) End = $ (35) So, the taget now can be depcted as whethe one can fnd two sets of scews whose ptches epesented by equaton (3) ae all zeos povded that they wee all ecpocal to $ End of equaton (35). Accodng to step, substtutng equaton (35) nto equaton (9) yelds the constants exeted to the end-effecto, $ End : End = whee ( ) $ = End $ End $ End $ End $ End $ End (36) $ epesents a foce along x -axs, $ = ( ) 3 epesents a foce along y -axs, $ = ( ) End End = 4 epesents a toque about x -axs, ( ) End = 5 axs, and ( ) End $ epesents a toque about y - $ epesents a toque about z-axs. om equaton (3), t s not dffcult to fnd that the sum of the numbe of the ndependent twsts and the numbe of the temnal constants of a chan s sx. In ode to educe the numbe of evolute jonts, one mght have to ncease the numbe of the temnal constants of the chans as many as possble. Accodng to equatons (3) and (36), the maxmum numbe of the temnal constants of a chan s fve. Howeve, f such a stuctue scheme s used, one may fnd each knematc chan only conssts of one evolute jont, whch s unfeasble n ealty. Smlaly, t s not dffcult to fnd that only when each knematc chan povdes thee temnal constants at most, can the stuctue scheme s feasble. Wth equaton (3), one can synthesze the temnal constants of these two knematc chans, ndvdually. Selectng dffeent (,,, 5) k = and substtutng them nto equaton (3), one can synthesze thee ndependent temnal constants fo the fst knematc chan, fo example: Assumng K =, one obtans 3 $ = $ $ $ (37)

21 486 Paallel Manpulatos, owads New Applcatons = = whee $ ( ) ndcates a foce along x -axs, $ ( ) = 3 ndcates a toque about y -axs, and ( ) axs. Assumng K = a b = b a, one can obtan $ ndcates a toque about z- 3 $ = $ $ $ (38) whee $ ( a b ) denotes a foce along the decton ( ) a b, $ = ( b a ) denotes a toque about the decton ( b a ) 3 = ( ) $ denotes a toque about z-axs and ab. Because span { K, K } 5 $ U= dm = must be equvalent to,, the esultant temnal constants of these knematc chans, $ End. So the constucton cteon s satsfed. Accodng to equaton (3), one mmedately obtans the twst bases fo the two knematc chans wth equatons (37) and (38): B whee ( ) = 3 $ = B $ B $ B $ B (39) $ epesents a otaton about x -axs, ( ) B $ epesents a tanslaton along y -axs, $ 3 ( ) = epesents a tanslaton along z-axs, and B = $ = $ $ $ (4) 3 B B B B B whee = ( cosα snα ) $ denotes a otaton about the decton ( cosα snα ), = ( snα cosα ) $ denotes a tanslaton along the B decton ( snα cosα ) 3, $ ( ) denotes a tanslaton along z -axs, B = and cosα = a and snα = b. a + b a + b Accodng to step 6, one can synthesze the twsts of the two knematc chans wth the twst bases (39) and (4), ndvdually. onsdeng the constucton cteon, one can fnd

22 Moblty of Spatal Paallel Manpulatos 487 that the least numbe of twsts n each knematc chan s thee. heefoe, the twst of the fst knematc chan can be syntheszed below wth equaton (34): ( c c c ) a 3 = c$ B + c$ B + c3$ B = 3 $ (4) Substtutng equaton (4) nto equaton (3) yelds: a h = (4) Equaton (4) ndcates that any twst havng the fom of equaton (4) wll natually satsfy the fee moton equements of the end-effecto. he atesan coodnates of the jont,, can be found fom equatons (7) and (9): A A s s = s a s + s = a c c 3 c c o make the twsts of the chan be equvalent to the twst bass, thee ae at least thee twsts ndcated n the fom of equaton (4). Suppose c = and the thee jonts coodnates ae ( a ) ( a yb zb ) ( a y z ) A = B = = then, the twsts of the fst knematc chan wll be: A = whee ( ) A B $ = AB $ $ $ (43) $ epesents a otaton about x -axs, ( z y ) B $ = epesents a otaton about a lne passng though pont B B ( x B yb zb ) and paallelng x -axs, and = ( z y ) otaton about a lne passng though pont ( y z ) $ epesents a x and paallelng x -axs. Accodng to equaton (34), a twst of the second knematc chan, denoted by expessed as: ( η cosα η snα η snα η α η ) a 3 B = η$ B + η$ B + η3$ B = cos 3 a $, can be $ (44) whee η denote eal numbes and =,, 3. Substtutng equaton (44) nto equaton (3) yelds: a h = (45)

23 488 Paallel Manpulatos, owads New Applcatons Equaton (45) ndcates that any twst havng the fom of equaton (44) wll natually satsfy the fee moton equements of the end-effecto. he atesan coodnates of the jont, A, can be found fom equatons (7) and (9): A s s = s b s η 3 + = snα + bcosα s η η3 cosα + bsnα η η η o keep the twsts of the chan be equvalent to the twst bass, one can only select thee ndependent twsts ndcated wth equaton (44) by selectng thee sets of ( η ) η. η 3 If one supposes η =, η =z, η3 = xsnα ycosα and b = x cosα + ysnα, he obtans the coodnates of evolute jont, = ( x y z ) ; smlaly, f one supposes η, η =z E, η = x snα y cosα and b = x E cosα + y snα, he obtans the coodnates of 3 E E evolute jont E, = ( x y z ) E E E E E ; and f one supposes η =, η =zd, η = x snα y cosα and b = x D cosα + y snα, he can obtan the coodnates of evolute 3 D D jont D, ( x yz ) D D D D D =. heefoe, the thee jonts coodnates can be assumed E D = = = ( x y z ) ( xe ye ze ) ( x y z ) then, the twsts of the second knematc chan wll be: D D D = E D $ = ED $ $ $ (46) = cos α sn α $ epesents a otaton about a lne passng though whee ( ) the ogn of the coodnate system and n the decton ( ) cosα snα, E = ( cosα snα zesnα zecosα xesnα yecosα ) passng though pont ( x y z ) and n the decton ( cos snα ) $ epesents a otaton about a lne E E D = ( cosα snα zdsnα zdcosα xdsnα ydcosα ) passng though pont ( x y z ) and n the decton ( cos snα ) E α, and $ epesents a otaton about a lne D D D α. Wth equatons (43) and (46), one can synthesze a spatal sx lnk mechansm ABDE shown n g.. It s not dffcult to fnd that α s the angle fom x -axs to the y' -axs of the evolute jont and the evolute jonts, E and D have the same axs decton, whch s denoted by = ( cosα snα ) n. ED

24 Moblty of Spatal Paallel Manpulatos 489 y y' z E D o α End-Effecto B A g. a Spatal Sx-Lnk Mechansm wth a Staght lne anslatonal End-Effecto om the above analyss, t s not dffcult to fnd that the two knematc chans AB and ED can suely guaantee the pue staght lne tanslaton of the end-effecto D so long as $ AB and $ ED do not descend n anks. o analyze the senstvty of the stuctue stablty to the angle α, one should tun to the equatons (37) and (38) and nvestgate the esultant temnal constants, whch can be expessed wth: whee cosα = a and snα = b. a + b a + b If the temnal constants denoted by ( α ) cosα snα $ D ( α ) = (47) snα cosα $ ae well condtoned, the mechansm wll D have fne stuctue stablty. om equatons (9) and (47), one can fnd that the end-effecto ank $ wll have one staght lne tanslaton along z-axs so long as ( α ) = 5 be mmedately tansfomed to nvestgate the followng sub matx of ( α ) x D D $ :, whch can

25 49 Paallel Manpulatos, owads New Applcatons Lettng det ( ( α )) = A ( α ) = cosα snα snα cosα A, one mmedately obtans α = o α = π. heefoe, n ode to keep the end-effecto D have one staght lne tanslaton along z-axs, thee wll be α and α π. So, the gd gudance mechansm syntheszed n ths chapte has a wde adaptaton of angle between the planes of ts two knematc chans. Now, the senstvty of the stuctue stablty to the angle α of the mechansm can be judged by the condton numbe of matx A ( α ) (Kelley, 995). Let cond λmax A A = λ mn A A ( A) = A A = (48) whee cond ( A) ndcates the condton numbe of matx A, matx A, and λ A A ndcates the egenvalues of matx A A. z y E D A ndcates the -nom of o End-Effecto B A g. a Spatal Sx-Lnk Mechansm wth the Best Stuctue Stablty he soluton of equaton (48) s: x π α = (49)

26 Moblty of Spatal Paallel Manpulatos 49 Equaton (49) ndcates that the mechansm wll have the best stuctue stablty when π α =, whch s shown n g.. ompaed wth Peaucelle-Lpkn eght-lnk lnkage, the spatal sx-lnk mechansm syntheszed n ths chapte has the least lnks and evolute jonts, and the whole end-effecto D can make an exact staght lne tanslaton whle Peaucelle- Lpkn eght-lnk lnkage can only allow one specfed pont to make such a moton. As a matte of fact, the mechansm shown n g. s a Saus lnkage. Howeve, the mechansm poposed hee does not necessaly eque that the two knematc chans must wthn two othogonal planes whch ae needed fo Saus lnkage. he so-called Saus lnkage, whch s shown n g., s a lnkage that convets ccula moton to lnea moton by usng hnged squaes. he squae end-effecto can make an exact staght lne tanslaton along z -axs whch shows bette popetes both n mechancal stuctue and n knematcs than those of Peaucelle-Lpkn eght-lnk lnkage. Howeve, because of the lmted wokspace and the uneconomc mechansm achtectue, the estctons of Saus lnkage shown n g. compaed wth the one shown n g. ae obvous. g. the Stuctue of Saus Lnkage z E D y o G B End-Effecto A g. 3 onfguaton of the Spatal Seven-Lnk Mechansm x

27 49 Paallel Manpulatos, owads New Applcatons As mentoned n step 6, n ode to keep the twsts of the chan to be equvalent to the twst bass, the ank of the twsts of each knematc chan syntheszed though equaton (34) should equal the dmenson of ts twst bass. heefoe, f one o moe such twsts ae added to each knematc chan, the fee motons of the end-effecto wll not be changed. As an example, the mechansm shown n g. 3 s the devatve fom of that n g. by addng one twst $ G to the knematc chan ED. Whee ( z x ) $ = G G G he two knematc chans of the end-effecto D ae now changed to be AB and he twsts of them two ae: EG D. A B $ = $ $ $ AB E G D $ = $ $ $ $ EGD It s not dffcult to fnd that the temnal constants of knematc chans AB and EG D ae stll expessed by equaton (36). And theefoe, the fee moton of the end-effecto D s stll a staght lne tanslaton along z-axs shown n g. 3. As a esult, the fee motons of the end-effecto wll not be changed f one o moe evolute jonts whose Plücke coodnates have the fom of equaton (44) ae added n the second knematc chan. Smlaly, the fee motons of the end-effecto wll not be changed ethe f one o moe evolute jonts whose Plücke coodnates have the fom of equaton (4) ae added to the fst knematc chan. Left: ont Suspenson g. 4 a ont Suspenson and a Rea Suspenson Rght: Rea Suspenson

28 Moblty of Spatal Paallel Manpulatos 493 o engneeng applcatons, the end-effecto D n g. o g. 3 can be utlzed as the gudng equpment of a mechansm that eques a pecse lnea tanslaton, such as the ndependent suspenson of automoble. Because the end-effecto of the gd gudance mechansm can make an exact staght lne tanslaton, the font and ea suspensons made up of such a mechansm shown n g. 4 allow the oentaton and poston paametes of the wheels such as kngpn, caste, cambe, and axes dstance and wheel tack to be constant. hese mets not only enhance the de and handlng of the vehcles, but also educe the weang of the tes dung jounce and ebound. 5. oncluson hs chapte focuses on the moblty analyss and synthess of spatal paallel manpulatos. It focuses on developng an analytcal methodology to nvestgate the nstantaneous DO of the end-effecto of a paallel manpulato and the nstantaneous contollablty of the endeffecto fom the vewpont of the possble actuaton schemes fo the paallel manpulato. Va compang the dffeences and essental moblty of a set of undeactuated, ove actuated and equally actuated manpulatos, ths chapte demonstates that the undeactuated, ove actuated and fully actuated manpulatos ae all substantally equally actuated mechansms. hs wok s sgnfcantly mpotant fo a desgne to contve hs o he manpulatos wth undeactuated o ove actuated stuctues. Based on the analytcal model of the DO of a spatal paallel manpulato, ths chapte also nvestgates a geneal pocess to synthesze the manpulatos wth specfed moblty. he outstandng chaactestcs of the synthess method ae that the whole pocess s also analytcal and each step can be pogammed at a compute. Because of the estctons of the tadtonal geneal moblty fomulas fo spatal mechansms, a lot of mechansms that mght not be syntheszed dectly wth the geneal moblty fomulas could be syntheszed wth ths analytcal theoy of degees of feedom fo spatal mechansms. 6. Acknowledgements hs eseach was suppoted by ANEDD unde Gant 74 and the Natonal Natual Scence oundaton of hna unde Gant he authos gatefully acknowledge these suppot agences. 7. Refeences Asenault, M. & Boudeau, R. (6). Synthess of Plana Paallel Mechansms Whle onsdeng Wokspace, Dextety, Stffness and Sngulaty Avodance, ASME Jounal of Mechancal Desgn, Vol. 8, No., (Januay, 6)69-78 ISSN: 5-47 Ball, R. S. (st Publshed 9, Repnted 998). a eatse on the heoy of Scews, ambdge Unvesty Pess, ISBN , ambdge Bglen, L. & Gosseln,. M. (6a). Geometc Desgn of hee-phalanx Undeactuated nges, ASME Jounal of Mechancal Desgn, Vol. 8, No., (Mach 6) , ISSN: 5-47

29 494 Paallel Manpulatos, owads New Applcatons Bglen, L. & Gosseln,. M. (6b). Gasp-state plane analyss of two-phalanx undeactuated fnges, Mechansm and Machne heoy, Vol. 4, No. 7, (July 6) 87-8, ISSN: 94-4X hakaboty, N. & Ghosal, A. (4). Knematcs of Wheeled Moble Robots on Uneven ean, Mechansm and Machne heoy, Vol. 39, No., (Decembe 4)73-87, ISSN: 94-4X Da, J. S. & Jones, J. Rees. (999). Moblty n Metamophc Mechansms of oldable/eectable Knds, ASME Jounal of Mechancal Desgn, Vol., No. 3, (999)375-38, ISSN: 5-47 Gogu, G. (5a). Moblty of Mechansms: a tcal Revew. Mechansm and Machne heoy, Vol. 4, No. 9, (Septembe 5) 68-97, ISSN: 94-4X Gogu, G. (5b). Moblty and Spatalty of Paallel Robots Revsted va heoy of Lnea ansfomatons, Euopean Jounal of Mechancs A/Solds, Vol. 4, No. 4, (July- August 5) 69-7, ISSN: Gosseln,. M. & Lalbeté,.. (998). Undeactuated Mechancal nge wth Retun Actuaton, Patent Numbe: 57639, Date of Patent: June 9, 998, Huang, Z.; Kong, L.. & ang, Y.. (997). heoy of Mechansm of Paallel Robotcs and ontol, Machney Industy Pess, ISBN: , Bejng Huang, Z.; Zhao, Y. S. & Zhao,. S. (6). Advanced Spatal Mechansm, Advanced Educatonal Pess of hna, ISBN: , Bejng Hunt, K. H. (978). Knematc Geomety of Mechansms, Oxfod Unvesty Pess, ISBN , Oxfod Jan, A. & Rodguez, G. (993) An Analyss of the Knematcs and Dynamcs of Undeactuated Manpulatos, IEEE ansactons on Robotcs and Automaton, Vol. 9, No. 4, (August 993)4-4, ISSN: 4-96X Kelley,.. (995). Iteatve Methods fo Lnea and Nonlnea Equatons, Noth aolna State Unvesty, Socety fo Industal and Appled Mathematcs, ISBN : (pbk.), Phladelpha Kempe, A. B. (877). How to Daw a Staght Lne, London: Macmllan. ted by Hendeson, D. W. & amna, D., Kuma, V. (99). Instantaneous Knematcs of Paallel-han Robotc Mechansms, ASME Jounal of Mechancal Desgn, Vol. 4, (Septembe 99) , ISSN: Lalbeté,. & Gosseln,. M. (998). Smulaton and Desgn of Undeactuated Mechancal Hands, Mechansm and Machne heoy, Vol. 33, No. -, (Januay-ebuay 998)39-57, ISSN: 94-4X Muakam,. et al. (989). Development of a new mult-lnk font suspenson, SAE Phllps, J. (984). eedom n Machney, Vol. : Intoducng Scew heoy, ambdge Unvesty Pess, ISBN , ambdge Phllps, J. (99). eedom n Machney, Vol. : Scew heoy Exemplfed, ambdge Unvesty Pess, ISBN , ambdge

30 Moblty of Spatal Paallel Manpulatos 495 Phllps, J. R. & Hunt, K. H. (964). On the heoem of hee Axes n the Spatal Moton of hee Bodes, Austalan Jounal of Appled Scence, Vol. 5, (964) 67-87, ISSN: Raghavan, M. (996). Numbe and Dmensonal Synthess of Independent Suspenson Mechansms, Mechansm and Machne heoy, Vol. 3, No. 8, (Novembe, 996)4-53, ISSN: 94-4X Raghavan, M. (5). Suspenson Synthess fo N: Roll ente Moton, ASME Jounal of Mechancal Desgn, Vol. 7, No. 4, (July 5) , ISSN: 5-47 Rotman, Joseph J. (). Advanced Moden Algeba, Pentce Hall, (May ) ISBN : Shgley, J. E. & Ucke, J. J. (98). heoy of Machnes and Mechansms, New Yok: McGaw-Hll ompanes, Inc., ISBN: , New Yok Smonescu, P. A. & Beale, D. (). Synthess and Analyss of the ve-lnk Rea Suspenson System Used n Automobles, Mechansm and Machne heoy, Vol. 37, No. 9, (Septembe, )85-83, ISSN: 94-4X Suh,. H. (989). Synthess and analyss of suspenson mechansms wth use of dsplacement matces, SAE sa, L.-W. (). Mechansm Desgn: Enumeaton of Knematc Stuctues Accodng to uncton, R Pess LL, ISBN , loda Waldon, K. J. (966). he onstant Analyss of Mechansms, Jounal of Mechansms, Vol., (966)-4, ISSN: 94-4X Yan, H.-S. & Kuo,.-H. (6). opologcal Repesentatons and haactestcs of Vaable Knematc Jonts, ASME Jounal of Mechancal Desgn, Vol. 8, No., (Mach, 6)384-39, ISSN: 5-47 Y, B.-J. & Km, W. K. (). he Knematcs fo Redundantly Actuated Omndectonal Moble Robots, Jounal of Robotc Systems, Vol. 9, No. 6, (June )55 67, ISSN: 74-3 Yu, Y. K. & L, Z. (3). ajectoy Geneaton fo a -dof Ove-actuated Paallel Manpulato wth Actuato Speed and oque Lmts onsdeaton, Poceedngs 3 IEEE Intenatonal Symposum on omputatonal Intellgence n Robotcs and Automaton, pp.58-63, ISBN: , July 6-, 3, Kobe, Japan. Zhao, J.-S. et al. (4a). A New Method to Study the Degee of eedom of Spatal Paallel Mechansms. he Intenatonal Jounal of Advanced Manufactung echnology, Vol. 3, No. 3-4, (ebuay 4) 88-94, ISSN: Zhao, J.-S.; Zhou, K. & eng, Z.-J. (4b). A heoy of Degees of eedom o Mechansms, Mechansm and Machne heoy, Vol. 39, No. 6, (June 4)6-643, ISSN: 94-4X Zhao, J.-S. et al. (6a). Re-analyss of the Degee-of-eedom onfguaton of the Platfoms n Spatal Paallel Mechansms wth onstants Spaces, he Intenatonal Jounal of Advanced Manufactung echnology, Vol. 8, No. -, (ebuay 6)9-96, ISSN: Zhao, J.-S. et al. (6b). he ee Moblty of a Paallel Manpulato, Robotca, Vol.4, No.5, (Septembe 6)635-64, ISSN:

31 496 Paallel Manpulatos, owads New Applcatons Zhao, J.-S.; eng, Z.-J. & Dong, J.- X. (6c). omputaton of the onfguaton Degee of eedom of a Spatal Paallel Mechansm by Usng Recpocal Scew heoy, Mechansm and Machne heoy, Vol. 4, No., (Decembe 6)486-54, ISSN: 94-4X

32 Paallel Manpulatos, towads New Applcatons Edted by Huapeng Wu ISBN Had cove, 56 pages Publshe I-ech Educaton and Publshng Publshed onlne, Apl, 8 Publshed n pnt edton Apl, 8 In ecent yeas, paallel knematcs mechansms have attacted a lot of attenton fom the academc and ndustal communtes due to potental applcatons not only as obot manpulatos but also as machne tools. Geneally, the ctea used to compae the pefomance of tadtonal seal obots and paallel obots ae the wokspace, the ato between the payload and the obot mass, accuacy, and dynamc behavou. In addton to the educed couplng effect between jonts, paallel obots bng the benefts of much hghe payload-obot mass atos, supeo accuacy and geate stffness; qualtes whch lead to bette dynamc pefomance. he man dawback wth paallel obots s the elatvely small wokspace. A geat deal of eseach on paallel obots has been caed out woldwde, and a lage numbe of paallel mechansm systems have been bult fo vaous applcatons, such as emote handlng, machne tools, medcal obots, smulatos, mco-obots, and humanod obots. hs book opens a wndow to exceptonal eseach and development wok on paallel mechansms contbuted by authos fom aound the wold. hough ths wndow the eade can get a good vew of cuent paallel obot eseach and applcatons. How to efeence In ode to coectly efeence ths scholaly wok, feel fee to copy and paste the followng: Jng-Shan Zhao ule hu and Zh-Jng eng (8). Moblty of Spatal Paallel Manpulatos, Paallel Manpulatos, towads New Applcatons, Huapeng Wu (Ed.), ISBN: , Inech, Avalable fom: l_manpulatos Inech Euope Unvesty ampus SeP R Slavka Kautzeka 83/A 5 Rjeka, oata Phone: +385 (5) ax: +385 (5) Inech hna Unt 45, Offce Block, Hotel Equatoal Shangha No.65, Yan An Road (West), Shangha, 4, hna Phone: ax: